Dear Folks I realize in replying to this I surely reach the end of possible comments that I can make for a week. But nevertheless … I want to comment on Terrence Deacon’s remarks below and also on Professor Johnstone’s remark from another email:
"This may look like a silly peculiarity of spoken language, one best ignored in formal logic, but it is ultimately what is wrong with the Gödel sentence that plays a key role in Gödel’s Incompleteness Theorem. That sentence is a string of symbols deemed well-formed according to the formation rules of the system used by Gödel, but which, on the intended interpretation of the system, is ambiguous: the sentence has two different interpretations, a self-referential truth-evaluation that is neither true nor false or a true statement about that self-referential statement. In such a system, Gödel’s conclusion holds. However, it is a mistake to conclude that no possible formalization of Arithmetic can be complete. In a formal system that distinguishes between the two possible readings of the Gödel sentence (an operation that would considerably complicate the system), such would no longer be the case. ********” I will begin with the paragraph above. Many mathematicians felt on first seeing Goedel’s argument that it was a trick, a sentence like the Liar Sentence that had no real mathematical relevance. This however is not true, but would require a lot more work than I would take in this email to be convincing. Actually the crux of the Goedel Theorem is in the fact that a formal system that can handle basic number theory and is based on a finite alphabet, has only a countable number of facts about the integers that it can produce. One can convince oneself on general grounds that there are indeed an uncountable number of true facts about the integers. A given formal system can only produce a countable number of such facts and so is incomplete. This is the short version of Goedel’s Theorem. Goedel worked hard to produce a specific statement that could not be proved by the given formal system, but the incompleteness actually follows from the deep richness of the integers as opposed to the more superficial reach of any given formal system. Mathematicians should not be perturbed by this incompleteness. Mathematics is paved with many formal systems. In my previous email I point to the Goldstein sequence. https://en.wikipedia.org/wiki/Goodstein%27s_theorem <https://en.wikipedia.org/wiki/Goodstein's_theorem> This is an easily understood recursive sequence of numbers that no matter how you start it, always ends at zero after some number of iterations. This Theorem about the Goodstein recursion is not provable in Peano Arithmetic, the usual formalization of integer arithmetic, using standard mathematical induction. This is a good example of a theorem that is not just a “Liar Paradox” and shows that Peano Arithmetic is incomplete. And by the way, the Goodstein sequence CAN be proved to terminate by using ‘imaginary values’ as Professor Deacon describes (with a tip of the hat to Spencer-Brown). In this case the imaginary values are a segment of Cantor’s transfinite ordinals. Once these transfinite numbers are admitted into the discussion there is an elegant proof available for the termination of the Goodstein sequence. Spencer-Brown liked to talk about the possibility of proofs by using “imaginary Boolean values”. Well, the Goodstein proof is an excellent example of this philosophy. A further comment, thinking about i (i^2 = -1) as an oscillation is very very fruitful from my point of view and I could bend your ear on that for a long time. Here is a recent paper of mine on that subject. Start in Section 2 if you want to start with the mathematics of the matter. http://arxiv.org/pdf/1406.1929.pdf <http://arxiv.org/pdf/1406.1929.pdf> And here is an older venture on the same theme. http://homepages.math.uic.edu/~kauffman/SignAndSpace.pdf <http://homepages.math.uic.edu/~kauffman/SignAndSpace.pdf> More generally, the idea is that one significant way to move out of paradox is to move into a state of time. I feel that this is philosophically a deep remark on the nature of time and that i as an oscillation is the right underlying mathematical metaphor for time. It is, in this regard, not an accident that the Minkowski metric is X^2 + Y^2 + Z^2 + (iT)^2. TIME = iT This is an equation with double meaning. Time is measured oscillation. Time is rotated ninety degrees from Space. And one can wonder: How does i come to multiply itself and return -1? Try finding your own answers before you try mine or all the previous stories! Best, Lou (See you next week.) > On May 2, 2016, at 9:31 PM, Terrence W. DEACON <dea...@berkeley.edu> wrote: > > A number of commentators, including the philosopher-logician G. Spencer > Brown and the anthropologist-systems theorist Gregory Bateson, reframed > variants of the Liar’s paradox as it might apply to real world phenomena. > Instead of being stymied by the undecidability of the logic or the semantic > ambiguity, they focused on the very process of analyzing these relationships. > The reason these forms lead to undecidable results is that each time they are > interpreted it changes the context in which they must be interpreted, and so > one must inevitably alternate between true and false, included and excluded, > consistent and inconsistent, etc. So, although there is no fixed logical, > thus synchronic, status of the matter, the process of following these > implicit injunctions results in a predictable pattern across time. In logic, > the statement “if true, then false” is a contradiction. In space and time, > “if on, then off” is an oscillation. Gregory Bateson likened this to a simple > electric buzzer, such as the bell in old ringer telephones. The basic design > involves a circuit that includes an electromagnet which when supplied with > current attracts a metal bar which pulls it away from an electric contact > that thereby breaks the circuit cutting off the electricity to the > electromagnet which allows the metal bar to spring back into position where > the electric contact re-closes the circuit re-energizing the electromagnet, > and so on. The resulting on-off-on-off activity is what produces a buzzing > sound, or if attached to a small mallet can repeatedly ring a bell. > Consider another variant of incompletability: the concept of imaginary > number. The classic formulation involves trying to determine the square root > of a negative number. The relationship of this to the liar’s paradox and the > buzzer can be illustrated by stepping through stages of solving the equation > i x i = -1. Dividing both sides by i produces i = -1/i, and then substituting > the value of i one gets i = -1/-1/i and then again i = -1/-1/-1/i and so > forth, indefinitely. With each substitution the value alternates from > negative to positive and cannot be resolved (like the true/false of the > liar’s paradox and the on/off of the buzzer). But if we ignore this > irresolvability and just explore the properties of this representation of an > irresolvable value, as have mathematicians for centuries, it can be shown > that i can be treated as a form of unity and subject to all the same > mathematical principles as can 1 and all the real numbers derived from it. So > i + i = 2i and i - 2i = -i and so on. Interestingly, 0 x i = 0 X 1 = 0, so we > can conceive of the real number line and the imaginary number line as two > dimensions intersecting at 0, the origin. Ignoring the many uses of such a > relationship (such as the use of complex numbers with a real and imaginary > component) we can see that this also has an open-ended consequence. This is > because the very same logic can be used with respect to the imaginary number > line. We can thus assign j x j = -i to generate a third dimension that is > orthogonal to the first two and also intersecting at the origin. Indeed, this > can be done again and again, without completion; increasing dimensionality > without end (though by convention we can at any point restrict this operation > in order to use multiple levels of imaginaries for a particular application, > there is no intrinsic principal forcing such a restriction). > One could, of course, introduce a rule that simply restricts such > operations altogether, somewhat parallel to Bertrand Russell’s proposed > restriction on logical type violation. But mathematicians have discovered > that the concept of imaginary number is remarkably useful, without which some > of the most powerful mathematical tools would never have been discovered. > And, similarly, we could discount Gödel’s discovery because we can’t see how > it makes sense in some interpretations of semiosis. On the other hand, like > G. Spencer Brown, Doug Hofstadter, and many others, thinking outside of the > box a bit when considering these apparent dilemmas might lead to other useful > insights. So I’m not so willing to brand the Liar, Gödel, and all of their > kin as useless nonsense. It’s not a bug, it’s a feature. > > > > On Mon, May 2, 2016 at 2:19 PM, Maxine Sheets-Johnstone <m...@uoregon.edu > <mailto:m...@uoregon.edu>> wrote: > Many thanks for your comments, Lou and Bruno. I read and pondered, > and finally concluded that the paths taken by each of you exceed > my competencies. I subsequently sent your comments to Professor > Johnstone—-I trust this is acceptable—asking him if he would care to > respond with a brief sketch of the reasoning undergirding his critique, > which remains anchored in Gödel’s theorem, not in the writings of others > about Gödel’s theorem. Herewith his reply: > > ******** > Since no one commented on the reasoning supporting the conclusions reached > in the two cited articles, let me attempt to sketch the crux of the case > presented. > > The Liar Paradox contains an important lesson about meaning. A statement that > says of itself that it is false, gives rise to a paradox: if true, it must be > false, and if false, it must be true. Something has to be amiss here. In > fact, what is wrong is the statement in question is not a statement at all; > it is a pseudo-statement, something that looks like a statement but is > incomplete or vacuous. Like the pseudo-statement that merely says of itself > that it is true, it says nothing. Since such self-referential > truth-evaluations say nothing, they are neither true nor false. Indeed, the > predicates ‘true’ and ‘false’ can only be meaningfully applied to what is > already a meaningful whole, one that already says something. > > The so-called Strengthened Liar Paradox features a pseudo-statement that says > of itself that it is neither true nor false. It is paradoxical in that it > apparently says something that is true while saying that what it says it is > not true. However, the paradox dissolves when one realizes that it says > something that is apparently true only because it is neither true nor false. > However, if it is neither true nor false, it is consequently not a statement, > and hence it says nothing. Since it says nothing, it cannot say something > that is true. The reason why it appears to say something true is that one and > the same string of words may be used to make either of two declarations, one > a pseudo-statement, the other a true statement, depending on how the words > refer. > > Consider the following example. Suppose we give the name ‘Joe’ to what I am > saying, and what I am saying is that Joe is neither true nor false. When I > say it, it is a pseudo-statement that is neither true nor false; when you say > it, it is a statement that is true. The sentence leads a double life, as it > were, in that it may be used to make two different statements depending on > who says it. A similar situation can also arise with a Liar sentence: if the > liar says that what he says is false, then he is saying nothing; if I say > that what he says is false, then I am making a false statement about his > pseudo-statement. > > This may look like a silly peculiarity of spoken language, one best ignored > in formal logic, but it is ultimately what is wrong with the Gödel sentence > that plays a key role in Gödel’s Incompleteness Theorem. That sentence is a > string of symbols deemed well-formed according to the formation rules of the > system used by Gödel, but which, on the intended interpretation of the > system, is ambiguous: the sentence has two different interpretations, a > self-referential truth-evaluation that is neither true nor false or a true > statement about that self-referential statement. In such a system, Gödel’s > conclusion holds. However, it is a mistake to conclude that no possible > formalization of Arithmetic can be complete. In a formal system that > distinguishes between the two possible readings of the Gödel sentence (an > operation that would considerably complicate the system), such would no > longer be the case. > ******** > > Cheers, > Maxine > _______________________________________________ > Fis mailing list > Fis@listas.unizar.es <mailto:Fis@listas.unizar.es> > http://listas.unizar.es/cgi-bin/mailman/listinfo/fis > <http://listas.unizar.es/cgi-bin/mailman/listinfo/fis> > > > > -- > Professor Terrence W. Deacon > University of California, Berkeley > _______________________________________________ > Fis mailing list > Fis@listas.unizar.es > http://listas.unizar.es/cgi-bin/mailman/listinfo/fis
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